Symmetry breaking at high temperatures in large N gauge theories

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Published for SISSA by Springer

Received: July 1, 2021 Accepted: August 5, 2021 Published: August 26, 2021

Symmetry breaking at high temperatures in large N gauge theories

Soumyadeep Chaudhuri and Eliezer Rabinovici Racah Institute, The Hebrew University,

Jerusalem 9190401, Israel

E-mail: chaudhurisoumyadeep@gmail.com,eliezer@mail.huji.ac.il

Abstract:Considering marginally relevant and relevant deformations of the weakly cou- pled (3 + 1)-dimensional large N conformal gauge theories introduced in [1], we study the patterns of phase transitions in these systems that lead to a symmetry-broken phase in the high temperature limit. These deformations involve only the scalar fields in the models.

The marginally relevant deformations are obtained by varying certain double trace quartic couplings between the scalar fields. The relevant deformations, on the other hand, are ob- tained by adding masses to the scalar fields while keeping all the couplings frozen at their fixed point values. At the N → ∞ limit, the RG flows triggered by these deformations approach the aforementioned weakly coupled CFTs in the UV regime. These UV fixed points lie on a conformal manifold with the shape of a circle in the space of couplings.

As shown in [1], in certain parameter regimes a subset of points on this manifold exhibits thermal order characterized by the spontaneous breaking of a global Z2 or U(1) symmetry and Higgsing of a subset of gauge bosons at all nonzero temperatures. We show that the RG flows triggered by the marginally relevant deformations lead to a weakly coupled IR fixed point which lacks the thermal order. Thus, the systems defined by these RG flows undergo a transition from a disordered phase at low temperatures to an ordered phase at high temperatures. This provides examples of both inverse symmetry breaking and sym- metry nonrestoration. For the relevant deformations, we demonstrate that a variety of phase transitions are possible depending on the signs and magnitudes of the squares of the masses added to the scalar fields. Using thermal perturbation theory, we derive the ap- proximate values of the critical temperatures for all these phase transitions. All the results are obtained at the N → ∞ limit. Most of them are found in a reliable weak coupling regime and for others we present qualitative arguments.

Keywords: Spontaneous Symmetry Breaking, Thermal Field Theory, 1/N Expansion ArXiv ePrint: 2106.11323

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Contents

1 Introduction 1

2 Review of the double bifundamental models 4

3 Phase transitions for marginally relevant deformations 8 3.1 Deformations of the double trace quartic couplings 8 3.2 RG flows triggered by the marginally relevant deformations 10

3.3 Estimates of the critical temperatures 11

4 Adding masses to the scalar fields 16

4.1 Case 1: M₁² 6= 0, M₂² = 0 19

4.1.1 Subcase 1: M₁²>0 19

4.1.2 Subcase 2: M₁²<0 21

4.2 Case 2: M₁² = 0, M₂² 6= 0 23

4.2.1 Subcase 1: M₂²>0 23

4.2.2 Subcase 2: M₂²<0 23

4.3 Summary of the results and comments on the caseM₁² 6= 0, M₂²6= 0 26

5 Conclusion and discussion 28

A Positivity of the quartic terms in the thermal effective potential at lead-

ing order 31

B RG flows of the masses of the scalar fields 32

1 Introduction

Spontaneous breaking of symmetries plays an important role in distinguishing the different phases of matter. Usually symmetries that are broken at low temperatures are eventually restored as the temperature becomes sufficiently high.¹ This is true also for systems in which a symmetry that is unbroken at low temperatures is spontaneously broken at a higher critical temperature [2] - a phenomenon called inverse symmetry breaking. In each such instance, the broken symmetry is restored at an even higher temperature. The ubiquity of such symmetry restoration in nature raised the question of whether there can be models where some symmetry remains broken up to arbitrarily high temperatures.

1Here, as well as in the rest of the paper, we are referring to the spontaneous breaking of ordinary (0-form) global symmetries. There are known instances of spontaneous breaking of higher form symmetries at high temperatures. One familiar example of this is the spontaneous breaking of the 1-form ZN center symmetry in pure Yang-Mills theory with SU(N) gauge group.

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This question has been explored for several decades in varied contexts; it has important implications for several areas of physics. We refer the reader to [3–14] for a sample of the literature. The main difficulty of coming up with a conclusive answer lay in the fact that the theories considered were UV-incomplete, thereby putting an upper cutoff on the temperatures that can be explored.² To overcome this problem certain Wilson-Fisher-like conformal field theories (CFTs) with globalO(N) symmetries were studied for both infinite and finite N in fractional dimensions [28, 29].³ The advantage of studying CFTs is that the absence of any intrinsic scale in the theory leads the system to be in the same phase (whatever it may be) at all nonzero temperatures. A symmetry-broken phase was indeed found to occur in the models considered in [28,29]. This was followed by the construction of (3+1)-dimensional large N conformal gauge theories in [1] where certain global Z2 or U(1) symmetries were shown to be broken at all nonzero temperatures. It was also shown that this symmetry breaking is accompanied by the Higgsing of a subset of gauge bosons which leads the system to be in a persistent Brout-Englert-Higgs (BEH) phase. In this context, we refer the reader to [31] for examples of similar persistent Higgsing at high temperatures in some asymptotically free large N gauge theories. Certain asymptotically safe theories [32] were also considered in [31], but they failed to show a symmetry-broken phase in the high temperature limit.

The study of the above-mentioned CFTs has demonstrated the existence of persistent symmetry breaking in the largeN limit. However, this still leaves open the question of how the persistent symmetry breaking characteristics change in the presence of an extra scale.

In this work we address this question by considering relevant and marginally relevant deformations of the large N CFTs introduced in [1]. These CFTs are weakly coupled.

They lie on a circle in the space of couplings. In appropriate parameter regimes, a subset of points on this fixed circle demonstrates spontaneous breaking of some globalZ2 or U(1) symmetries at any nonzero temperature. We consider two kinds of deformations of these CFTs with thermal order. As we will discuss shortly, these deformations involve only the scalar fields in the model. At the N → ∞ limit, the RG flows triggered by these deformations end up at the aforementioned CFTs in the UV regime. So, in this limit these systems provide examples of symmetry nonrestoration in theories with nontrivial UV fixed points. Exploring the IR regimes of these flows, we find interesting patterns of phase transitions at nonzero temperatures. Let us briefly describe these deformations and the corresponding phase transitions in the following two paragraphs.

The first class of deformations that we consider consists of marginally relevant ones involving variations of certain quartic couplings between the scalar fields. Such marginally relevant deformations may exist in four dimensional theories only in the presence of non-

2Examples of symmetry nonrestoration have been found in certain models with imaginary or random chemical potentials [15–20]. The dynamics of these models suffer from the lack of unitarity. Some holo- graphic models were also explored as candidates for theories with symmetry nonrestoration [21–27]. Unlike the phases discussed in this paper, the symmetry-broken phases in these models do not correspond to stable vacua in thermal states.

3There may be issues with unitarity of such Wilson-Fisher-like fixed points at finite N due to the potential existence of operators with complex scaling dimensions [30].

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Abelian gauge interactions [33]. This is the case here. We show that the RG flows corre- sponding to these deformations lead to a non-Gaussian fixed point in the IR. The theory remains weakly coupled throughout the flow allowing the use of perturbation theory to study it. As mentioned earlier, under certain conditions, some of the UV fixed points of these flows demonstrate persistent thermal order. On the other hand, the IR fixed point lacks thermal order. From this we conclude that there must be an inverse symmetry breaking phase transition at an intermediate temperature. By a perturbative analysis, we determine the critical temperature corresponding to this phase transition. Above the critical temperature, the system persistently remains in a symmetry-broken phase.

The second class of deformations involves adding masses to the scalar fields while keeping all the couplings frozen at the fixed points exhibiting thermal order. We show that at high temperatures, the effects of these masses are insignificant and the system exists in the same phase as the UV fixed point where a Z2 (or U(1)) global symmetry remains persistently broken. As the temperature is decreased the effects of the renormalized masses of the scalar fields become increasingly pronounced. At sufficiently low temperatures, such renormalized masses can induce phase transitions in the system. We show that in certain cases such phase transitions can be studied using thermal perturbation theory. We derive estimates of the critical temperatures corresponding to these phase transitions. We show that a variety of phases can exist slightly below such critical temperatures. The nature of these phases crucially depend on the magnitudes and signs of the squares of the renormalized masses. Far below the critical temperatures, the perturbative analysis breaks down. So we cannot say anything definite about the phases in this regime.

We emphasize that all the results mentioned above are derived in the N → ∞ limit.

Whether the symmetry nonrestoration in these models persists for finite N remains unre- solved.⁴ In the conclusion of the paper we comment on the potential problems that may arise for symmetry nonrestoration at finite N.

Organization of the paper: in section 2, we review the models that were introduced in [1]. We discuss the features of the planar beta functions of the couplings in these models and the corresponding fixed points with special emphasis on the fixed points with thermal order.

In section 3, we demonstrate the existence of marginally relevant deformations of the fixed points exhibiting thermal order. We show that the systems defined by these defor- mations undergo inverse symmetry breaking phase transitions at nonzero critical temper- atures. We derive estimates of these critical temperatures.

In section 4, we add masses to the scalar fields in the models, and study the phase transitions induced by these masses at nonzero critical temperatures. We show that there are distinct patterns of phase transitions for the different signs and magnitudes of the squares of these masses. We provide estimates of the critical temperatures corresponding to these phase transitions.

4In this context, we refer the reader to the recent work [34] where symmetry nonrestoration was found to occur even at finiteN in somed-dimensional nonlocal CFTs with 1< d <4.

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In section 5, we conclude by summarizing our results and commenting on how finite N corrections can alter the features found in the N → ∞ limit.

In appendix A we show that the tree-level quartic terms in the effective potential of the scalar fields are positive-definite throughout the RG flows discussed in section 3.

In appendix B we derive the 1-loop beta functions of the masses of the scalar fields that are introduced in section4.

2 Review of the double bifundamental models

In this section we will describe the models that were originally introduced in [1] and which will be the objects of interest in this work. These models have gauge groups of the form

G=

2

Y

i=1

G_i×G_i (2.1)

where Gi can be either SO(Nci) or SU(Nci).⁵ The models where the Gi’s are SO(Nci) and SU(N_ci) were called the real double bifundamental model and the complex double bifundamental model respectively in [1]. These names were coined keeping in mind the representations in which certain scalar fields in the model transform under the gauge group.

Henceforth, we will use the abbreviations RDB and CDB for these models. The two sectors of the gauge group are labeled by the index i. The matter fields in each sector transform only under the gauge group (G_i×G_i) in that sector and they are invariant under the gauge transformations in the other sector. The fields in thei^th sector include two sets of massless fermions (ψi and χi), each of which has Nf i flavors and transforms in the fundamental representation of one of the G_i’s while being invariant under the otherG_i. These fermions are Majorana spinors in the RDB model and Dirac spinors in CDB model. In addition to these fermions, there is an Nci×Nci matrix of massless scalar fields which transform in the bifundamental representation of G_i×G_i. These scalars are real in the RDB model and complex in the CDB model. The scalar fields in each sector interact via both single trace and double trace quartic couplings. There is an additional double trace interaction coupling the scalar fields in the two sectors. In figure 1 we provide a schematic diagram indicating the representations in which the different fields transform in these models.

The renormalized Lagrangians of these models have the following common form:⁶ L=−1

2

2

X

i,γ=1

Tr^h(Fiγ)µν(Fiγ)^µνi+iκ

2

X

i=1

Tr^hψ_iDψ/ i

i+iκ

2

X

i=1

Tr^hχ_iDχ/ i

i

+κ

2

X

i=1

Tr^hDµΦi

^† D^µΦi

i−

2

X

i=1

ehiTr^h(Φ^†_iΦi)²ⁱ−

2

X

i=1

fei

Tr^hΦ^†_iΦi

i2

−2ζ^eTr^hΦ^†1Φ1

iTr^hΦ^†2Φ2

i,

(2.2)

whereκ= ¹₂ for the RDB model, andκ= 1 for the CDB model. The index γ distinguishes the twoGi’s in the i^th sector. The fermionic fieldsψi andχi are (Nci×N_{f i}) matrices.

5The ranksNc1 andNc2can be different.

6In case of the RDB model, Φ^†_i = Φ^T_i since the scalar fields are real.

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Nc1

Nc1

Nc2

Nc2

Nf1

Nf1

Nf2

Nf2

Nf2

Φ1 Φ2

ψ1

χ₁

ψ2

χ₂

Figure 1. A schematic diagram indicating the representations in which the different fields trans- form in the double bifundamental models: the two sectors are represented by the two subdiagrams which are connected by a dashed line. This dashed line represents a double trace quartic interaction between the scalar fields in the two sectors. The two green nodes in each sector represent the two G_i’s in the gauge groupG_i×G_i of that sector. Each of theseG_i’s is SO(N_ci) for the RDB model and SU(N_ci) for the CDB model. The line connecting the two green nodes in each sector represents the scalar fields in that sector which transform in the bifundamental representation of Gi×Gi. These scalar fields interact via both single trace and double trace quartic couplings. The two yellow nodes in each sector represent theN_{f i}flavors of the two fermionsψ_iandχ_iin that sector. Each line connecting a green node and a yellow node indicates that the respective fermion is an (Nci×Nf i) matrix which transforms in the fundamental representation of the corresponding Gi.

In [1] the fixed points of these models were studied at the Veneziano limit [35] where N_ci, N_{f i} → ∞ while r ≡ ^N_N^c2

c1 and x_{f i} ≡ ^N_N^{f i}

ci are kept finite, and the different couplings scale as

g_i² = 16π²λi

N_ci , ^eh_i= 16π²hi

N_ci , f^e_i= 16π²fi

N_ci² , ζ^e= 16π²ζ

N_c1N_c2 (2.3) withgi being the gauge coupling in the i^th sector.⁷ We will continue to work in this limit.

In this limit, there is an orbifold equivalence [36–45] between the RDB model and the CDB model [1] with the couplings in the two dual theories related by

λ^C_i = λ^R_i

2 , h^C_i = 2h^R_i , f_i^C = 2f_i^R, ζ^C = 2ζ^R. (2.4) Here the superscript ‘R’ or ‘C’ indicates whether the coupling belongs to the RDB or the CDB model. This planar equivalence between the two models will allow us to restrict our attention to the RDB model. All results that will be derived in this work will have their counterparts in the CDB model.

Now, the planar beta functions (in the MS scheme) of the different couplings in the RDB model have the following forms:

β_λi =−

21−4x_{f i} 6

λ²_i +−27 + 13x_{f i} 6

λ³_i +· · · , β_hi = 16h²_i −6hiλi+ 3

16λ²_i +· · ·, β_fi = 8f_i²+ 32f_ih_i−6f_iλ_i+ 24h²_i + 9

16λ²_i + 8ζ²+· · ·, β_ζ =ζ^h8f₁+ 8f₂+ 16h₁+ 16h₂−3λ₁−3λ₂ⁱ+· · · .

(2.5)

7The gauge couplings for the twoGi’s in thei^th sector are taken to be equal.

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The dots indicate higher order corrections. There are several features of these beta func- tions which will be important for our analysis in this work. We would like to mention two of them here. Firstly, note that these beta functions are independent of the ratio r= ^N_N^c2_c1. This means that the RG flows that we will study in section 3 will be independent of this ratio. Secondly, the beta functions of the single trace couplings (λi and hi) are indepen- dent of the double trace couplings (f_i and ζ).⁸ These properties of the beta functions were shown to survive up to all orders at the Veneziano limit in [1].

Based on the forms of the above beta functions, one can show that there are unitary perturbative fixed points⁹ which are of the following two kinds.

1. There is a discrete set of fixed points where the two sectors are decoupled and the couplings have the following values at leading order:

λi = 21−4x_{f i}

13xf i−27, hi= 3−√ 6

16 λi, fi = 2√ 6 +σi

q18√ 6−39

16 λi, ζ = 0 (2.6) with σi = ±1. Note that to get unitary fixed points in a perturbative regime, one must setx_{f i}= ²¹₄ −i with 0< i 1.

2. The second class of fixed points consists of theories where the two sectors are coupled.

These fixed points exist only whenx_f1 =x_f2 ≡x_f. They form a conformal manifold which has the shape of a circle in the space of couplings.¹⁰ The values of the different couplings on this fixed circle at leading order are as follows:

λ₁ =λ₂≡λ= 21−4x_f

13x_f−27, h₁=h₂ ≡h= 3−√ 6 16 λ, f_p =

√6

8 λ, f_m² +ζ²=18√ 6−39 256

λ².

(2.7)

wherefp ≡ ^f1+f₂ ² and fm≡ ^f1−f₂ ².

All these fixed points survive under higher loop corrections at the planar limit and have stable effective potentials (at least up to leading order in perturbation theory) [1]. More- over, in appropriate regimes of the ratio rcertain points on the fixed circle exhibit thermal order, i.e., aZ2symmetry in the sector with the smaller rank is spontaneously broken at all nonzero temperatures. The relevantZ2 symmetry in thei^thsector transforms the different fields in that sector as follows:

Φi→ T_iΦi, ψ_i → T_iψ_i, (V_i1)µ→ T_i(V_i1)µT_i⁻¹, (2.8) whereT_i is the followingNci×Nci diagonal matrix:

T_i ≡diag{−1,1,· · ·,1}. (2.9)

8This is a general feature of planar beta functions of single trace couplings in large N gauge theories [39].

9These fixed points are akin to the Banks-Zaks-Caswell fixed points [46–48] in more familiar QCDs.

10A couple of points on this circle whereζ= 0 have already been included in the first class.

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It leaves the fields in the other sector unchanged. These symmetry transformations map one class of gauge-equivalent field configurations to another.¹¹ These global symmetries were called baryon symmetries in [1] because the corresponding order parameters are the expectation values of the determinants of the scalar fields.

When Nc2 < Nc1, the baryon symmetry in the first sector remains unbroken at all temperatures. On the other hand, the baryon symmetry in the second sector can be spontaneously broken in a thermal state for a subset of points on the conformal manifold when r < r_max≡

r

6√ 6−13 61−6√

6 ≈0.191. These points can be expressed simply by defining the following polar coordinates in the fm-ζ plane:

fm =Rsinθ, ζ =Rcosθ, (2.10)

where θ ∈[0,2π). For each r < rmax, the fixed points exhibiting thermal order lie in the angular intervalθ∈(θ₁, θ₂) where¹²

θ₁= cos⁻¹(ν₁), θ₂=π−sgn(r−r₀)π−cos⁻¹(ν₂) (2.11) withr₀≡

√

18√ 6−39

12 and

ν1≡r





−12 +^q 18√

6−183r²+ 18√

6−39 1 +r²

q18√

16−39



 ,

ν2≡r





−12−^q 18√

6−183r²+ 18√

6−39 1 +r²

q18√

16−39



 .

(2.12)

Here the range of the function cos⁻¹ is the interval [0, π]. The plots of θ₁ and θ₂ as functions of r are shown in figure 2. From these plots, one can see that for all values of r ∈(0, rmax) the end points of the above angular interval satisfy ^π₂ < θ1 < θ2 < ^3π₂ with θ₁→ ^π₂ and θ₂ → ^3π₂ asr→0, andθ_1,2→cos⁻¹−

√

61−6√ 6 4√

3

≈2.952 asr→r_max. Thus, cosθ (or equivalently, ζ = Rcosθ) is always negative for the fixed points demonstrating thermal order.

When N_c1 < N_c2, one would get similar fixed points with spontaneous breaking of the baryon symmetry in the first sector at all nonzero temperatures. These fixed points can be obtained from those demonstrating thermal order in the second sector by the transfor- mations r→ ¹_r andθ→2π−θ.

It was shown in [1] that for all the fixed points where the baryon symmetry in a sector is spontanously broken at nonzero temperatures, this phenomenon is accompanied by the Higgsing of half of the gauge bosons in the same sector. This leads the system to be in a persistent Brout-Englert-Higgs (BEH) phase at all nonzero temperatures.

A similar spontaneous breaking of a baryon symmetry at nonzero temperatures and a persistent BEH phase were also found at the corresponding fixed points of the dual CDB

11See appendix A of [1] for a proof of this.

12One can check that the angular interval given here is equivalent to the intervals offmandζgiven in [1].

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0.00 0.05 0.10 0.15 r

2.0 2.5 3.0 3.5 4.0 4.5

θ

θ1

θ2

Figure 2. Graphs of θ1 and θ2 againstr: for each value of r < rmax ≡q

6√ 6−13 61−6√

6, θ1 and θ2 are the endpoints of the angular interval on the fixed circle which exhibits thermal order in the second sector. They always lie in the regime ^π₂ < θ1 < θ2 < ^3π₂. As r →0, θ1 → ^π₂ and θ2 → ^3π₂. As r→rmax,θ1,2→cos⁻¹

−

√

61−6√ 6 4√

3

≈2.952.

model. In this case, the relevant baryon symmetry in each sector is a U(1) symmetry which can be obtained by replacingT_i by the Nci×Nci diagonal matrix defined below:

(T_i)φ≡diag{e^iφ,1,· · · ,1}. (2.13) withφ∈[0,2π).

An important result which was derived in [1] is that the above-mentioned baryon symmetries are not broken in a thermal state for the fixed points where the two sectors are decoupled. We will find this result to be consequential as one of these fixed points is the IR limit of the RG flows corresponding to the marginally relevant deformations discussed in section 3.

3 Phase transitions for marginally relevant deformations

In this section we will consider certain marginally relevant deformations of the points on the fixed circle in the RDB model by varying the quartic couplings between the scalars fields. In subsection 3.1 we will demonstrate the existence of such a marginally relevant deformation for each point on the fixed circle. Later in subsection3.2we will study the RG flows triggered by these deformations. We will show that in the IR limit these RG flows take the corresponding systems to a unique fixed point that lacks thermal order. Therefore, at low temperatures all these systems are in a disordered phase. On the other hand, in the UV limit some of these systems flow to the points on the fixed circle which exhibit thermal order. This means that at high temperatures these systems are in an ordered phase. Therefore, each of these systems must undergo an inverse symmetry breaking phase transition at some critical temperature. Above this critical temperature, the system remains persistently in a symmetry-broken phase. In subsection 3.3, we will provide an estimate of this critical temperature.

3.1 Deformations of the double trace quartic couplings

Let us now turn our attention to the deformations of the quartic couplings away from their values on the fixed circle. The deformations that are of interest to us involve only the double

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trace couplings (f_i and ζ). Since the beta functions of the single trace couplings (λ_i and hi) are independent of the double trace ones at the Veneziano limit, these couplings remain frozen at their fixed point values throughout the RG flows triggered by such deformations.

Therefore, all such RG flows take place only in a subspace where λ₁ = λ₂ = λ and h1 =h2 =h withλ andh being the values given in (2.7) at leading order in perturbation theory. In this subspace the full planar beta functions of the double trace couplings have the following forms:¹³

β_fp =F₁f_p,^qf_m² +ζ², β_fm =f_mF^e₂f_p,^qf_m² +ζ², β_ζ =ζF^e₂f_p,^qf_m² +ζ², (3.1) whereF₁ andF^e₂ are two functions off_p and^pf_m² +ζ². Switching to the polar coordinates introduced in (2.10), the beta functions can be re-expressed as

β_fp =F₁(f_p, R), β_R=F₂(f_p, R), Rβ_θ = 0, (3.2) whereF2(fp, R)≡RF^e2(fp, R). The expressions ofF1(fp, R) andF2(fp, R) at leading order are

F₁(f_p, R) = (8f_p+ 32h−6λ)f_p+ 8R²+ 24h²+ 9 16λ², F2(fp, R) =R(16fp+ 32h−6λ) .

(3.3) We can see that βθ = 0 for all points in the space of the double trace couplings where R6= 0. Moreover, the beta functions offpandRare independent of the angular coordinate θ. Therefore, for a fixed point of these beta functions where R6= 0, a shift in the value of θ spans the conformal manifold. Another important consequence of the vanishing of βθ is that the projections of all RG flows on the f_m-ζ plane are radially directed. Furthermore, the fact that β_fp and β_R are independent of θensures that such flows are identical for all values ofθ.

Now, suppose f_0p and R₀ are the values of f_p and R for the points on the conformal manifold. The leading order expressions of f_0p and R₀ are (see (2.7))

f0p =

√6

8 λ, R0 =

q18√ 6−39 16

!

λ. (3.4)

Consider the following small deformations of fp and R away from their values on the fixed circle:

fp =f0p+δfp, R=R0+δR. (3.5) The beta functions of these deformations in the values of f_p andR (at linear order) are

β_δfp = ∂F₁

∂fp(f_0p, R₀)δf_p+ ∂F₁

∂R(f_0p, R₀)δR, β_δR= ∂F2

∂f_p(f0p, R0)δfp+ ∂F2

∂R(f0p, R0)δR.

(3.6)

13We refer the reader to section 5 of [1] for a derivation of these forms of the beta functions of the double trace couplings up to all orders in perturbation theory at the planar limit.

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The coefficient matrix multiplying the deformations in the above equations is

M=





∂F1

∂fp(f_0p, R₀) ^∂F_∂R¹(f_0p, R₀)

∂F2

∂fp(f_0p, R₀) ^∂F_∂R²(f_0p, R₀)



= 16R₀ 0 1 1 0

!

. (3.7)

The second equality in the above equation is obtained by substituting F₁, F₂ and the couplings h,f0p and R0 by their leading order expressions. The eigenvalues of this matrix and the corresponding eigenvectors are as follows:

v1=−16R0, v2 = 16R0, e1= 1

−1

!

, e2 = 1 1

! .

(3.8)

Therefore, the deformations alonge₁ and e₂ are marginally relevant and marginally irrele- vant respectively. The existence of a marginally relevant deformation clearly demonstrates that all points on the conformal manifold are UV fixed points of certain RG flows. Next, let us study the features of these RG flows.

3.2 RG flows triggered by the marginally relevant deformations

To analyze the RG flows triggered by the marginally relevant deformations, we will rely on the 1-loop expressions of the beta functions. This approximation is justified near the UV fixed points on the conformal manifold as these fixed points are weakly coupled. However, the RG flows triggered by the marginally relevant deformations can take the system to a strongly coupled regime. We will show that this is not the case when we choose the marginally relevant deformation such that it has a radially inward component in the fm-ζ plane, i.e.,δR <0. Therefore, in this case one can trust the results based on the 1-loop beta functions. If the deformation is chosen to be in the opposite direction, i.e., δR >0, then the theory indeed flows to a strongly coupled regime. This can be verified by noticing that the RG flows obtained from the 1-loop beta functions lead to divergences of the couplings at finite energy scales.

With the above comments in mind, we choose the marginally relevant deformation to be δf_p=−δR= R₀

2 ^e (3.9)

at a reference energy scale Λ. We take 0 <_e < 1. The RG flow generated by the 1-loop beta functions leads to the following values of the couplings at a scale µ:

f_p=f_0p+R₀ 2

"

(1−k) 1−tanh(8R₀t) 1 +ktanh(8R₀t)

#

, R= R₀ 2

"

(1 +k) 1 + tanh(8R₀t) 1 +ktanh(8R₀t)

#

, (3.10) where t ≡ ln(µ/Λ) and k ≡ 1−_e. In figure 3, a graphical plot of this RG flow is given forλ= 0.001, _e= 0.1. Note that the theory remains weakly coupled throughout the RG flow. In the deep UV, it flows to a point on the fixed circle as expected. In the deep IR, it flows to a fixed point where the two sectors are decoupled and f₁ =f₂ = ^2√6+

√

18√ 6−39

16 λ.

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-6000 -4000 -2000 2000 4000 t

0.00032 0.00034 0.00036 0.00038 0.00040 0.00042 0.00044

fp

-6000 -4000 -2000 2000 4000 t

0.00002 0.00004 0.00006 0.00008 0.00010 0.00012 0.00014

R

Figure 3. RG flow forλ= 0.001, e= 0.1: in the UV limit (t→ ∞) the couplings flow to a point on the fixed circle. In the IR limit (t→ −∞) they flow to a fixed point where the two sectors are decoupled.

Due to the rotational symmetry in thef_m-ζ plane, this behavior of the RG flow is identical for marginally relevant deformations of all points on the fixed circle. In particular, when r < rmax, it holds for the deformations of the fixed points demonstrating thermal order in the second sector. The RG flows triggered by these deformations lead to the IR fixed point where the two sectors are decoupled. As we have mentioned earlier, the baryon symmetries in both the sectors are unbroken at any nonzero temperature for this IR fixed point. Therefore, for the systems that flow from the UV fixed points with thermal order to this IR fixed point without thermal order, there must be a transition from a disordered phase to an ordered phase as the temperature is increased. We will now determine the critical temperatures at which this phase transition takes place in these systems.

3.3 Estimates of the critical temperatures

To determine the critical temperatures, we will consider the thermal effective potential of the scalar fields. At leading order in perturbation theory, this potential has both quadratic and quartic terms. The quartic terms are as follows:

V_quartic = 16π²

" ₂ X

i=1

h_i

N_ciTr[(Φ^Ti Φi)²] +

2

X

i=1

f_i N_ci²

Tr[Φ^Ti Φi]²+ 2ζ

N_c1N_c2Tr[Φ^T1Φ1]Tr[Φ^T2Φ2]

# . (3.11) In appendixAwe have shown that these quartic terms are positive-definite throughout the RG flows that we have discussed in the previous subsection.

The quadratic terms in the thermal effective potential are generated due to integration over the nonzero Matsubara modes of the different fields in the theory. These terms have the following structure

V_quadratic= 1 2

2

X

i=1

m²_th,iTr[Φ^T_i Φi] (3.12) where m²_th,1 and m²_th,2 are the thermal masses (squared) of the scalar fields Φ1 and Φ2

respectively. The contributions of 1-loop diagrams to such thermal masses (squared) were computed for a general 4-dimensional gauge theory in [3]. Using these general results, the 1-loop expressions of m²_th,1 and m²_th,2 were derived in [1]. These expressions reduce to the

JHEP08(2021)148

following forms at the planar limit:

m²_th,1= 16π²T² 3

2h₁+f₁+rζ+3 8λ₁

, m²_th,2= 16π²T²

3

2h₂+f₂+ζ r +3

8λ₂

,

(3.13)

whereT is the temperature. These thermal masses (squared) quantify the behavior of the effective potential of the scalar fields in the neighborhood of the point Φ1 = Φ2 = 0, viz., the origin of the field space. Their signs determine the fates of the baryon symmetries at any given temperature. If either m²_th,1 or m²_th,2 is negative then the minimum of the effective potential cannot be at the origin of the field space. In fact, the positivity of the tree-level quartic terms in the potential ensures the existence of a minimum of the potential away from the origin. This would imply the spontaneous breaking of at least one of the baryon symmetries. In particular, when m²_th,i<0, the baryon symmetry in the i^th sector would be spontaneously broken [1].

To track the effective potential at different temperatures we will set the renormalization scale µ=T. Let us explain this choice of the renormalization scale. Our aim is to check perturbatively whether the thermal effective potential of the scalar fields has a minimum away from the origin. When both the thermal masses (squared) are positive, the minimum of the potential is at the origin. In this case, as long as we are sufficiently away from the critical temperature, we will have m²_th,1, m²_th,2 ∼ ^16π₃²λT².¹⁴ At higher orders in the perturbative expansion of the potential near the origin, there are terms which contain the logarithm of the ratio of m²_th,i and µ². The lowest order term of this kind is the Coleman-Weinberg term [49–51]. By setting µ=T we ensure that such logarithmic terms are suppressed by at least a factor of λlnλ 1 compared to the leading order terms that we are retaining.¹⁵ A similar argument can be given when either of the thermal masses (squared) is negative and the minimum of the potential is away from the origin.

In this case, again as long as we are sufficiently away from the critical point, we will have |m²_th,i| ∼ ^16π₃²λT² irrespective of the sign of m²_th,i.¹⁶ Comparing this behavior of the thermal masses (squared) with the quartic terms given in (3.11), one can check that if there is symmetry-breaking in the i^th sector, then the value of Φi (appropriately normalized by Nci) at the minimum would be of the same order as the temperature scale T. To be more precise, if (Φi)0 is the value of Φi at the minimum, then

√

Tr[(Φi)^T₀(Φi)0]

Nci ∼T.¹⁷ We want the perturbative expansion of the potential to be valid at this scale. At higher orders in this expansion, there are terms which contain the logarithm of the ratio of λ times the square of this scale and µ². By taking µ=T, we again ensure that such logarithmic corrections

14This is due to the fact that the different couplings are at mostO(λ) throughout the RG flows discussed in the previous subsection. From the expressions given in (3.13), one can see that ifris too small or too large then the magnitude of one of the thermal masses (squared) can become much larger than ^16π₃²λT². In this paper, we will work in a regime where this is not the case.

15This suppression persists even whenT is very small.

16In fact, this behavior of the thermal masses (squared) would persist in the high temperature limit.

17See [1] for the explicit forms of such thermal expectation values of the scalar fields in the UV CFTs.

Similar expressions would hold in the models that we are studying presently.

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are small. So we can rely on the perturbative analysis sufficiently away from the critical temperature on both sides of this temperature. To come up with an estimate of the critical temperature, we will interpolate the thermal masses (squared) in the intermediate interval with the expressions given in (3.13) and search for the point at which one of the thermal masses (squared) vanishes.

One may be able to improve the above-mentioned perturbative analysis by choosing µ to be someO(1) numerical coefficient timesT. Such a numerical factor can be absorbed in the reference energy scale Λ by rescaling it appropriately while doing thermal perturbation theory. Due to the ambiguity in the value of this numerical factor, there would be theoret- ical uncertainties in the critical temperatures of the phase transitions that we will study.¹⁸ Henceforth, we will ignore this subtlety as our aim is to find the qualitative nature of the phases in different temperature regimes and to determine the critical temperatures for the phase transitions in terms of the unspecified reference energy scale Λ.

Having provided the rationale for choosing µ=T in the effective potential, let us now look at the behavior of the thermal masses at different temperature scales. For this it is convenient to define the dimensionless quantity

me²_th,i≡ 3

16π²T²m²_th,i. (3.14)

Substituting the values of the different couplings along the RG flow triggered by the marginally relevant deformation, we get

me²_th,1 =R0

2 (1 +k)rcosθ+ sinθ−1C(t) +3 4λ+R0

, me²_th,2 =R0

2 (1 +k)cosθ

r −sinθ−1C(t) +3 4λ+R0

,

(3.15)

whereθ is the angular location of the UV fixed point on the conformal manifold andC(t) is the following function oft≡ln(^T_Λ):

C(t)≡ 1 + tanh(8R₀t)

1 +ktanh(8R₀t). (3.16)

Note that the parameterk= 1−_econtrols the behavior of the thermal masses at different temperatures. As e → 0, C(t) → 1 and the rescaled thermal masses (squared), me²_th,1 and me²_th,2, stop changing with the temperature. In this limit, they just reduce to the rescaled thermal masses (squared) of the points on the conformal manifold which were analyzed in [1].

To study the variations of the thermal masses (squared) with the temperature con- cretely, we choose the UV fixed point to be the point where θ = π, or equivalently, fm = 0, ζ = −R₀. As can be verified from (2.11), this fixed point has thermal order in the second sector only when r < r₀ ≈0.188. So to stay in this regime, we chooser = 0.1.

To keep the values of the couplings and the deformations small we choose λ= 0.001 and e = 0.1 as before. For this set of parameters, the plots of the rescaled thermal masses (squared) at different temperature scales are given in figure 4. As one can see form these

18Such theoretical uncertainties were discussed in [52] for first order phase transitions in some models.

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-6000 -4000 -2000 2000 4000 t=ln(T/Λ)

0.00075 0.00080 0.00085 0.00090

m²_th,1

-6000 -4000 -2000 2000 4000 t=ln(T/Λ)

-0.0005 0.0005

m²th,2

Figure 4. Plots of the rescaled thermal masses (squared) against t= ln(T /Λ) for r= 0.1,θ=π, λ= 0.001,e= 0.1: in the low temperature regime, bothme²_th,1andme²_th,2saturate at positive values.

This indicates that the baryon symmetries in both the sectors are unbroken in this regime. In the high temperature limit,me²_th,1saturates at a positive value, whileme²_th,2saturates at a negative value.

This means that the baryon symmetry in the second sector remains persistently broken above a critical temperature. This critical temperature (T_c) is given byt_c≡ln(T_c/Λ)≈ −1172.1.

plots, bothme²_th,1 andme²_th,2decrease monotonically with increase in the temperature. How- ever, me²_th,1 remains positive at all temperatures indicating the absence of thermal order in the first sector. On the other hand, me²_th,2 starts off from a positive value at low tem- peratures, but eventually becomes negative at high temperatures indicating a transition to an ordered phase. The critical temperature at which this transition happens is where me²_th,2 = 0. This behavior is similar for all the systems where the UV fixed point exhibits thermal order in the second sector. One can obtain a general expression of the critical temperature (Tc ≡ Λe^tc) for all these systems by solving the equation me²_th,2 = 0. We provide this expression below:

tc= 1

8R0 tanh⁻¹(ρ), (3.17)

where

ρ≡ −(1 +k)^R_2r⁰ −cosθ+r(1 + sinθ)+ (^3λ₄ +R0)

(1 +k)^R_2r⁰ −cosθ+r(1 + sinθ)−k(^3λ₄ +R₀). (3.18) From the above expression, we can see that t_c has a real value only whenρ∈(−1,1). This puts a restriction on the UV fixed points for which there can be a phase transition. The allowed fixed points are precisely the ones whereθ∈(θ1, θ2) withθ1andθ2being the values given in (2.11). To study the variation oft_c in this domain, we choose r= 0.1, λ= 0.001, e = 0.1 and plot the values of tc for different values of θ in figure 5. As one can see, tc

increases rapidly as one approaches the edges of the interval (θ₁, θ₂). Asθ→θ₁ orθ→θ₂, t_c → ∞thereby indicating the absence of the phase transition at the end points. For each point in the interval θ ∈ (θ1, θ2), there is a finite temperature at which the system goes to an ordered phase characterized by the spontaneous breaking of the baryon symmetry and Higgsing of half of the gauge bosons in the second sector. In the Veneziano limit, the system remains in this phase at all higher temperatures.

The line of critical points shown in figure 5separates the following two phases:

• Phase 1: the high temperature phase where the baryon symmetry in the first sec- tor remains unbroken while the baryon symmetry in the second sector is sponta- neously broken.